Modelling the Charging of a Capacitor

I consulted some text books, which gave me what I attached as a screenshot, for dv/dt of a capacitor from the time it is just to be about charged. To cut a long story short, I'm suppose to model it using the analytical equation and Euler's method. However the voltage DE I got from the text book itsn't working out. It says:

(dv/dt) = (v - Vs) / RC,
where v is the voltage across the cap, and Vs is the applied voltage. Using this equation makes me end up with a neg, or just wrong voltage as opposed to the corrent analytical equation of